三类非线性椭圆和抛物方程奇异解的渐近性态与稳定性分析

In this project, we will study the asymptotic behaviour and stability of singular solutions, gradient blowup solutions and global solutions for a class of semilinear elliptic equations and two kinds of parabolic equations with nonlinear gradient source. The problems come from certain stellar dynamic models and an appropriate model for surface growth by ballistic deposition, and specifically for vapour deposition and the sputter deposition of thin films of aluminium and rare earth metals respectively. Due to the essential difficulty in estimates of solutions by singlarity, nonlinear gradient source or nonlinear operator, there are some strong challenges in mathematical theory. Here, combining mathematical analysis and numerical simulation, we mainly consider several difficult problems related the mathematical theory. By some new ideas and techniques, we analyze the exactly asymptotic behaviors of singular and gradient blowup solutions, the stability of their behavior and the large time behaviors of global solutions. Furthermore, we want to indicate the influence of singularity, nonlinear gradient source or operator. These questions in this project have more important value in the mathematical theory, and the related subjects are also the frontier and popular topics in the field today of reaction-diffusion equations or nonlinear evolution equations.

本项目拟对一类半线性椭圆型方程和两类含非线性梯度的抛物型方程奇异解、梯度爆破解和全局解的渐近性态与稳定性进行研究。该问题分别源自天体物理学中星球运动模型和物理力学中矿藏的表面增长、蒸汽沉积，以及铝合金等金属的薄膜成形研究。由于方程的奇性、非线性梯度或算子给相应的解估计带来了本质理论困难，数学理论的研究极具挑战性。项目主要围绕数学理论上的若干难点问题开展研究，采用理论分析结合数值模拟，分析奇异解和梯度爆破解精确的渐近性态，分析解的对应性态的稳定性，分析全局解的大时间性态，探讨奇异性和非线性梯度或算子对解的性态的影响。项目研究具有较高的数学理论价值，相关课题的研究也是目前反应扩散或非线性发展方程研究领域中前沿和热门的研究课题。

本项目对一类半线性椭圆型方程和两类含非线性梯度的抛物型方程奇异解、梯度爆破解和全局解的渐近性态与稳定性进行了系统的研究。该问题分别源自天体物理学中星球运动模型和物理力学中矿藏的表面增长、蒸汽沉积，以及铝合金等金属的薄膜成形研究。项目围绕数学理论上的若干难点问题开展，结合数值模拟，主要进行理论分析，研究了奇异解和梯度爆破解精确的渐近性态，研究了全局解、熄灭解、最大模爆破解和梯度爆破解的完整分类，研究了全局解或奇异解的大时间和小参数收敛性，深入探讨了算子、反应项梯度和指数、以及初值，对解的渐近性态的影响。项目共发表高水平的国际期刊SCI论文11篇，国际重要期刊论文2篇，中文重要期刊论文1篇，培养1名博士生和3名硕士生毕业，指导7名在读博士生。圆满完成了项目预定的研究目标和任务，形成了一套较为系统和有效的理论和方法。